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OPENCV – PYTHON | Geometric Transformations | Euclidean, Affine, Projective | Mathematics + CODE

Welcome to the enchanting world of image transforms, where OpenCV and Python come together to perform visual wizardry. In this journey, we’ll demystify the Euclidean, Affine, and Projective transforms, unveiling the mathematical intricacies that govern these pixel metamorphoses. Brace yourself for a captivating blend of theory and code, as we explore the transformative magic within OpenCV.

 

Flow of the Article

  1. Euclidean Transform with Code
  2. Affine Transform with Code
  3. Projective Transform with Code
  4. Major Difference
  5. Conclusion

 

Video Explanation of the Topic:

 

Euclidean Transform: A Dance of Rotation and Translation

Euclidean transforms gracefully orchestrate rotations and translations, infusing images with a delightful twist. In the realm of mathematics, the Euclidean transform matrix looks like this:

Here, θ represents the rotation angle, and (tx,ty) denotes the translation along the x and y axes. Let’s bring this mathematical marvel to life with OpenCV:

import cv2
import numpy as np

# Load an image
image = cv2.imread(‘path_to_your_image.jpg’)

# Define the Euclidean transform parameters
theta = 45 # rotation angle in degrees
tx, ty = 50, 30 # translation in x and y

# Create the Euclidean transform matrix
euclidean_matrix = np.float32([[np.cos(np.radians(theta)), -np.sin(np.radians(theta)), tx],
[np.sin(np.radians(theta)), np.cos(np.radians(theta)), ty]])

# Apply the Euclidean transform
euclidean_transformed_image = cv2.warpAffine(image, euclidean_matrix, (image.shape[1], image.shape[0]))

# Display the result
cv2.imshow(‘Euclidean Transformed Image’, euclidean_transformed_image)
cv2.waitKey(0)
cv2.destroyAllWindows()

Affine Transform: The Art of Preserving Parallelism

Affine transforms elevate the transformation game by preserving parallel lines, offering a more flexible manipulation of images. The affine transform matrix takes the form:

Here, a and d control scaling, b and c manage shear, and (tx,ty) handle translation. Behold the mathematical symphony translated into code:

import cv2
import numpy as np

# Load an image
image = cv2.imread(‘path_to_your_image.jpg’)

# Define the Affine transform parameters
a, b, c = 1.5, 0.5, 0.2 # scaling and shear
tx, ty = 50, 30 # translation in x and y

# Create the Affine transform matrix
affine_matrix = np.float32([[a, b, tx],
[c, d, ty]])

# Apply the Affine transform
affine_transformed_image = cv2.warpAffine(image, affine_matrix, (image.shape[1], image.shape[0]))

# Display the result
cv2.imshow(‘Affine Transformed Image’, affine_transformed_image)
cv2.waitKey(0)
cv2.destroyAllWindows()

Projective Transform: Warp and Weave

Projective transforms, or perspective transforms, add a touch of warp and weave to the visual narrative. The matrix for a projective transform takes a more intricate form:

These h coefficients form the heart of perspective warping. Let’s dive into code to witness the warp in action:

import cv2
import numpy as np

# Load an image
image = cv2.imread(‘path_to_your_image.jpg’)

# Define four source points (corners of a rectangular region)
src_points = np.float32([[56, 65], [368, 52], [28, 387], [389, 390]])

# Define four destination points
dst_points = np.float32([[0, 0], [300, 0], [0, 300], [300, 300]])

# Create the projective transform matrix
projective_matrix = cv2.getPerspectiveTransform(src_points, dst_points)

# Apply the projective transform
projective_transformed_image = cv2.warpPerspective(image, projective_matrix, (300, 300))

# Display the result
cv2.imshow(‘Projective Transformed Image’, projective_transformed_image)
cv2.waitKey(0)
cv2.destroyAllWindows()

Major Differences :

In the realm of image transformation, Euclidean, Affine, and Projective transforms each bring their unique flavor to the table. Euclidean transforms are like the elegant ballet dancers, gracefully spinning images through rotations and translations, while Affine transforms add a touch of flexibility, preserving parallel lines and introducing scaling and shear. Projective transforms, on the other hand, are the visual illusionists, warping perspectives and creating a sense of depth. In simple terms, Euclidean is about turning and shifting, Affine is about stretching and squeezing, and Projective is about playing visual tricks to transform images in ways that capture the imagination. These transformations, like different brushes in an artist’s toolkit, offer varied ways to reshape the canvas of pixels with OpenCV’s magic touch.

 

Top 10 Most Asked Questions and Answers:

Here are ten important interview questions related to image transformation using OpenCV, along with their answers:

  1. Question: What is the fundamental difference between Euclidean and Affine transforms?

    • Answer: Euclidean transforms involve only rotations and translations, preserving distances and angles. Affine transforms, in addition to rotations and translations, introduce scaling and shear while maintaining parallelism.

  2. Question: How is a rotation matrix constructed for an image in OpenCV?

    • Answer: A rotation matrix in OpenCV is created using the cv2.getRotationMatrix2D function, specifying the rotation center, angle, and scaling factor.

  3. Question: Explain the importance of the transformation matrix in image processing.

    • Answer: The transformation matrix encapsulates the parameters for image transforms, guiding OpenCV to perform operations such as rotation, translation, scaling, and shearing. Understanding and manipulating this matrix are crucial for effective image processing.

  4. Question: What role does interpolation play in image scaling?

    • Answer: Interpolation determines how pixel values are estimated when scaling an image. OpenCV provides various interpolation methods, such as linear and cubic, to maintain visual quality during resizing.

  5. Question: How does an Affine transform matrix differ from a Projective transform matrix?

    • Answer: An Affine transform matrix is a 2×3 matrix that includes parameters for scaling, rotation, and translation. In contrast, a Projective transform matrix is a 3×3 matrix that additionally accounts for perspective distortion.

  6. Question: Can you explain how to perform a Projective transform on an image using OpenCV?

    • Answer: The cv2.getPerspectiveTransform function is used to obtain the projective transform matrix, and cv2.warpPerspective applies this matrix to warp the image based on specified source and destination points.

  7. Question: How does shear differ from scaling in Affine transformations?

    • Answer: Scaling changes the size of the object uniformly, while shear skews the object along one axis, introducing deformation without changing its overall size.

  8. Question: Why might one choose a Projective transform over an Affine transform?

    • Answer: A Projective transform is suitable when dealing with perspective changes, such as correcting images taken from an angle. It allows for more complex distortions, making it a powerful tool for certain real-world scenarios.

  9. Question: How does the choice of interpolation method impact image quality during scaling?

    • Answer: The interpolation method affects how pixel values are estimated between existing pixels. While linear interpolation is faster, cubic interpolation provides smoother results, especially when upscaling images.

  10. Question: Can you describe a practical application where understanding image transformations is crucial?

    • Answer: Image transformations are vital in computer vision tasks, such as object recognition and augmented reality. For instance, correcting perspective distortion in images of documents is essential for accurate text recognition in OCR applications.